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Math Help - Series Divergence

  1. #1
    Senior Member
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    Feb 2008
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    Series Divergence

    I'm finding this question a bit difficult

    Let s_n denote the nth partial sum of the series:

    1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...

    1) Show that s_n>\sqrt{n} whenever n>1

    2) Hence explain why the series diverges.



    All I notice from this is that the series is:

    \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}

    And that this is a p-series which must obviously diverge
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  2. #2
    Member
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    Jan 2009
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    1. n,i\in Z^{+}~,~n>i\rightarrow \displaystyle\sqrt{\frac{n}{i}}>1\rightarrow \sqrt{n}\sum_{i=1}\frac{1}{\sqrt{i}}>n


    \rightarrow  \displaystyle S_n=\sum_{i=1}\frac{1}{\sqrt{i}}>\sqrt{n}

    2. \displaystyle S_n=\sum_{i=1}\frac{1}{\sqrt{i}}>\sum_{i=1}\frac{1  }{i} , so series diverges.
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